| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2019 |
| Session | Specimen |
| Marks | 4 |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Standard +0.3 This is a straightforward PGF question requiring expansion of the binomial, applying the condition that probabilities sum to 1 to find 'a', identifying possible values from non-zero coefficients, and using the standard derivative formula for E(X). All steps are routine applications of PGF theory with no novel insight required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general |
| Answer | Marks |
|---|---|
| \(E(X) = G_X'(1) = a(4t^3 + 6t + 0 - 2t^{-3}) | _{t=1}\) [M1] |
**Question 1(a)**
$G_X(t) = at\left(t + \frac{1}{t}\right)^3 = at(t^3 + 3t + 3t^{-1} + t^{-3}) = a(t^4 + 3t^2 + 3t^0 + t^{-2})$ [M1]
$X$ takes the values 4, 2, 0, −2 [A1]
$G_X(1) = 1$ *or* sum of coefficients = 1 [M1]
$\Rightarrow a = \frac{1}{8}$ [A1]
**Total: 4 marks**
**Question 1(b)**
$E(X) = G_X'(1) = a(4t^3 + 6t + 0 - 2t^{-3})|_{t=1}$ [M1]
Differentiate and evaluate at $t = 1$ **OR** by symmetry
$= 1$ [**ft** $8a$] [A1, A1ft]
**Total: 2 marks**1 The discrete random variable X has probability generating function $\mathrm { G } _ { X } ( t )$ given by
$$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in either order, the value of $a$ and the set of values that $X$ can take.
\item Find the value of $\mathrm { E } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2019 Q1 [4]}}